A shipping company uses a cargo plane to transport two types of cargo containers: Type A a… — SAT

A shipping company uses a cargo plane to transport two types of cargo containers: Type A and Type B. Let $x$ represent the number of Type A containers, and let $y$ represent the number of Type B containers. The constraints on the shipment are represented by the system of inequalities below:

\begin{aligned} x + y &\leq 45 \\ 3x + 5y &\leq 165 \end{aligned}

If the plane must carry at least $10$ Type A containers, what is the maximum number of Type B containers the plane can transport?

A
25
27Cevap
C
35
D
39

Cevap

The correct answer of 27 is found by substituting the minimum value of

x

, which is 10, into the second inequality constraint. This yields

3(10) + 5y \leq 165

, which simplifies to

5y \leq 135

, or

y \leq 27

. Since the coordinate pair (10, 27) also satisfies the first inequality constraint (

10 + 27 = 37 \leq 45

), 27 is the maximum possible value for

y

Adım Adım Çözüm

Identify the given constraints and the variable to maximize.

We are given the system of inequalities

x + y \leq 45

and

3x + 5y \leq 165

, with

x \geq 10

, and we want to find the maximum possible value of

y

This establishes the mathematical model and boundaries for the problem.

Express

y

in terms of

x

for both inequalities to see which constraint is more restrictive.

From the first inequality:

y \leq 45 - x

. From the second inequality:

3x + 5y \leq 165 \implies 5y \leq 165 - 3x \implies y \leq 33 - 0.6x

This allows us to analyze

y

as a function of

x

under both constraints.

Substitute the minimum possible value of

x

, which is 10, into both boundary expressions for

y

Under the first constraint:

y \leq 45 - 10 = 35

. Under the second constraint:

y \leq 33 - 0.6(10) = 27

Since both boundary lines have negative slopes, the maximum value of

y

will occur when

x

is at its minimum value of 10.

Determine the most restrictive upper bound for

y

when

x = 10

The value of

y

must satisfy both

y \leq 35

and

y \leq 27

, so the maximum possible value is 27.

A solution to a system of inequalities must satisfy all inequalities in the system simultaneously.

Anahtar Kavram

Solving systems of linear inequalities in context by evaluating boundary conditions and identifying constraints.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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